It doesn't mean "really good" just "good". While you can use this construction of "good at -ing" I think it sounds smoother to use the noun form of the action verb. You are a good dancer. You are a good writer. You are a bad singer. Etc.

Yes, you can say both of those - 'good at writing' or 'bad at writing'. You can also say good/bad at for other things that people do, for example, 'good at dancing' or 'bad at singing'.